Quadrics (from wikipedia)

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are curves. Such quadrics are the same as conic sections, and are typically known as conics rather than quadrics.

Ellipse (e=1/2), parabola (e=1) and hyperbola (e=2) with fixed focus F and directrix.

In Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. By making a suitable Euclidean change of variables, any quadric in Euclidean space can be put into a certain normal form by choosing as the coordinate directions the principal axes of the quadric. In three-dimensional Euclidean space there are 16 such normal forms.^[2] Of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all.^[3]

Non-degenerate real quadric surfaces
Ellipsoid	${x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,$
Spheroid (special case of ellipsoid)	${x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,$
Sphere (special case of spheroid)	${x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,$
Elliptic paraboloid	${x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,$
Circular paraboloid(special case of elliptic paraboloid)	${x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,$
Hyperbolic paraboloid	${x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,$
Hyperboloid of one sheet	${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,$
Hyperboloid of two sheets	${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,$
Degenerate quadric surfaces
Cone	${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,$
Circular Cone (special case of cone)	${x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 0 \,$
Elliptic cylinder	${x^2 \over a^2} + {y^2 \over b^2} = 1 \,$
Circular cylinder (special case of elliptic cylinder)	${x^2 \over a^2} + {y^2 \over a^2} = 1 \,$
Hyperbolic cylinder	${x^2 \over a^2} - {y^2 \over b^2} = 1 \,$
Parabolic cylinder	$x^2 + 2ay = 0 \,$